Inequality Problems in Mechanics and Applications

Abstract

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Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics

“…Furthermore, from (19) and (38) we obtain that u (t) − Div σ(t) = f 0 (t) in Q. This equality combined with (32) and (58) imply that Div σ ∈ L 2 (0, T ; V * ).…”

“…t ∈ (0, T ). Using this notation and inserting (37) into (38), Problem P V is equivalently formulated as follows.…”

“…The notion of a hemivariational inequality, a useful generalization of variational inequality, is concerned with nonconvex and nonsmooth energy functionals and was introduced and studied in the early 1980s by Panagiotopoulos in [36,38,39]. This type of inequality is based on the notion of the generalized directional derivative and the Clarke generalized gradient of a locally Lipschitz function and is closely related to a class of nonlinear inclusions of subdifferential type.…”

History-Dependent Problems with Applications to Contact Models for Elastic Beams

“…We claim that for a.e. t ∈ (0, T ) the solution w depends continuously on the right hand side g. Indeed, let g 1 , g 2 ∈ V * and w 1 , w 2 ∈ V be the corresponding solutions to (16). We have…”

“…Considerable progress has been achieved in their modeling, mathematical analysis and numerical simulations, and, as a result, a general Mathematical Theory of Contact Mechanics is currently emerging. It is concerned with the mathematical structures which underly general contact problems with different constitutive laws, i.e., materials, various geometries and different contact conditions, see for instance [7,8,14,16,17,19] and the references therein. An important number of contact problems arising in Mechanics, Physics and Engineering Science lead to mathematical models expressed in terms of subdifferential inclusions, and variational and hemivariational inequalities.…”